In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach spaces), and give a conceptual proof of the Weierstrass division theorem.
Lecture two covers smooth, unramified and étale morphisms between noetherian rings. The relation of these notions with the concepts of submersion, immersion and diffeomorphism from differential geometry is given.
In the third lecture, we investigate ring extensions between the three power series rings and describe the respective flatness properties. This allows us to prove approximation in the linear case.
The last lecture is devoted to the geometry of solution sets in power series spaces. We construct in the case of one $x$-variable an isomorphism of an $m$-adic neighborhood of a solution with the cartesian product of a (singular) scheme of finite type with an (infinite dimensional) smooth space, thus extending the factorization theorem of Grinberg-Kazhdan-Drinfeld.
CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université[-]

In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach spaces), and give a conceptual proof of the Weierstrass division theorem.
Lecture two covers smooth, unramified and étale morphisms between noetherian rings. The relation of these notions with the concepts of submersion, immersion and diffeomorphism from differential geometry is given.
In the third lecture, we investigate ring extensions between the three power series rings and describe the respective flatness properties. This allows us to prove approximation in the linear case.
The last lecture is devoted to the geometry of solution sets in power series spaces. We construct in the case of one $x$-variable an isomorphism of an $m$-adic neighborhood of a solution with the cartesian product of a (singular) scheme of finite type with an (infinite dimensional) smooth space, thus extending the factorization theorem of Grinberg-Kazhdan-Drinfeld.
CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université[-]

In this series of four lectures we develop the necessary background from commutative algebra to study solution sets of algebraic equations in power series rings. A good comprehension of the geometry of such sets should then yield in particular a "geometric" proof of the Artin approximation theorem.
In the first lecture, we review various power series rings (formal, convergent, algebraic), their topology ($m$-adic, resp. inductive limit of Banach spaces), and give a conceptual proof of the Weierstrass division theorem.
Lecture two covers smooth, unramified and étale morphisms between noetherian rings. The relation of these notions with the concepts of submersion, immersion and diffeomorphism from differential geometry is given.
In the third lecture, we investigate ring extensions between the three power series rings and describe the respective flatness properties. This allows us to prove approximation in the linear case.
The last lecture is devoted to the geometry of solution sets in power series spaces. We construct in the case of one $x$-variable an isomorphism of an $m$-adic neighborhood of a solution with the cartesian product of a (singular) scheme of finite type with an (infinite dimensional) smooth space, thus extending the factorization theorem of Grinberg-Kazhdan-Drinfeld.
CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université[-]

The $p$-adic Igusa zeta function, topological and motivic zeta function are (related) invariants of a polynomial $f$, reflecting the singularities of the hypersurface $f = 0$. The first one has a number theoretical flavor and is related to counting numbers of solutions of $f = 0$ over finite rings; the other two are more geometric in nature. The monodromy conjecture relates in a mysterious way these invariants to another singularity invariant of $f$, its local monodromy. We will discuss in this survey talk rationality issues for these zeta functions and the origins of the conjecture.[-]

One of the possible applications of Artin approximation is to prove that the local geometry of sets defined in affine space by real or complex analytic equations is not more complicated than the local geometry of sets defined by polynomial equations. A possible approach is to prove that a complex analytic (singular) germ, for example $(X,0) \subset (\mathbf{C} ^n,0)$, is the intersection, in some affine space $\mathbf{C}^N$, of an algebraic germ $(Z,0) \subset (\mathbf{C}^N,0)$ by a complex analytic non singular subspace $(W,0)$ of dimension $n$ which is "in general position" with respect to $Z$ at the origin. Approximating $Z$ by an algebraic subspace then yields the desired result, provided the "general position" condition is sufficiently precise. I will explain how one can attack this problem using a notion of "general position with respect to a singular space" which is based on the concept of minimal Whitney stratification, which will also be explained. Nested Artin approximation is essential in this approach.

nested Artin approximation - Whitney forms - singularities - stratifications - germ of subspace[-]

Let $A$ be the ring of formal power series in $n$ variables over a field $K$ of characteristic zero. Two power series $f$ and $g$ in $A$ are said to be equivalent if there exists a $K$-automorphism of $A$ transforming $f$ into $g$. In my talk I will review criteria for a power series to be equivalent to a power series which is a polynomial in at least some of the variables. For example, each power series in $A$ is equivalent to a polynomial in two variables whose coefficients are power series in $n - 2$ variables. In particular, each power series in two variables over $K$ is equivalent to a polynomial with coefficients in $K$. Similar results are valid for convergent power series, assuming that the field $K$ is endowed with an absolute value and is complete. In the special case of convergent power series over the field of real numbers some weaker notions of equivalence will be also considered. I will report on works of several mathematicians giving simple proofs. Some open problems will be included.

singularities - power series[-]

Michael ARTIN participated in the "Artin Approximation and Infinite dimensional Geometry" event organized at CIRM in March 2015, which was part of the Jean-Morlet semester held by Herwig Hauser. Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry and also generally recognized as one of the outstanding professors in his field. Artin was born in Hamburg, Germany, and brought up in Indiana. His parents were Natalia Jasny (Natascha) and Emil Artin, a preeminent algebraist of the 20th century. In 2002, Artin won the American Mathematical Society's annual Steele Prize for Lifetime Achievement. In 2005, he was awarded the Harvard Centennial Medal. He won the Wolf Prize in Mathematics. He is also a member of the National Academy of Sciences and a Fellow of the American Academy of Arts and Sciences, the American Association for the Advancement of Science, the Society for Industrial and Applied Mathematics, and the American Mathematical Society.[-]

Across many mathematical disciplines one encounters projective systems of algebraic varieties indexed by a combinatorial datum, such as a natural number, a finite graph, or a lattice polytope. As the datum grows, the algebraic complexity of the corresponding variety (measured, for instance, in terms of its defining equations or higher-order syzygies) typically increases. But in good cases it eventually stabilises in a well-defined manner, especially when the family admits a direct system of sufficiently large symmetry groups. Exactly when this stabilisation phenomenon can be expected is still poorly understood, and this question motivates much current research activity in algebraic geometry and adjacent branches. After a brief general setup, which involves passing to the projective limit of the varieties and the direct limit of their symmetry groups, I will discuss a number of concrete instances where stabilisation occurs, both from classical algebraic geometry and from other areas of mathematics.[-]

We will present some of the original definitions, results, and proof techniques about Pfaffian functions on the reals by Khovanskii.
A simple example of a Pfaffian function is an analytic function $f$ in one variable $x$ satisfying a differential equation $f^\prime = P(x,f)$ where $P$ is a polynomial in two variables. Khovanskii gives a notion of complexity of Pfaffian functions which in the example is just the degree of $P$. Using this complexity, he proves analogues of Bézout's theorem for Pfaffian curves (say, zero loci of Pfaffian functions in two variables), with explicit upper bounds in terms of the ocurring complexities.
We explain a recent application by J. Pila and others to a low-dimensional case of Wilkie's conjecture on rational points of bounded height on restricted Pfaffian curves. The result says that the number of rational points of height bounded by $T$, on a transcendental restricted Pfaffian curve, grows at most as a power of log$(T)$ as $T$ grows. This improves the typical upper bound $T^\epsilon$ in Pila-Wilkie's results in general o-minimal structures, the improvement being due to extra geometric Bézout-like control.
In the non-archimedean setting, I will explain analogues of some of these results and techniques, most of which are (emerging) work in progress with L. Lipshitz, F. Martin and A. Smeets. Some ideas in this case come from work by Denef and Lipshitz on variants of Artin approximation in the context of power series solution.[-]

The concept of a "transseries" is a natural extension of that of a Laurent series, allowing for exponential and logarithmic terms. Transseries were introduced in the 1980s by the analyst Écalle and also, independently, by the logicians Dahn and Göring. The germs of many naturally occurring real-valued functions of one variable have asymptotic expansions which are transseries. Since the late 1990s, van den Dries, van der Hoeven, and myself, have pursued a program to understand the algebraic and model-theoretic aspects of this intricate but fascinating mathematical object. A differential analogue of “henselianity" is central to this program. Last year we were able to make a significant step forward, and established a quantifier elimination theorem for the differential field of transseries in a natural language. My goal for this talk is to introduce transseries without prior knowledge of the subject, and to explain our recent work.[-]